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Alberto Farina

Publications and source records attributed to Alberto Farina.

At least 37 records · Page 2Linked to original sources
arXiv2020

Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature

In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $u$ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $u$.

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arXiv2019

Moving planes for domain walls in a coupled system

The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized to hyperfine spin states with a Rabi term coupling. This leads to domain wall solutions having a monotone structure for a non-cooperative system. We use the moving plane method to prove mono-tonicity and one-dimensionality of the phase transition solutions. This relies on totally new estimates for a type of system for which no Maximum Principle a priori holds. We also derive that one dimensional solutions are unique up to translations. When the Rabi coefficient is large, we prove that no non-constant solutions can exist.

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arXiv2019

A Liouville theorem for the $p$-Laplacian and related questions

We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.

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arXiv2019

Pointwise gradient bounds for entire solutions of elliptic equations with non-standard growth conditions and general nonlinearities

We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.

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arXiv2018

Bernstein-Moser-type results for nonlocal minimal graphs

We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs. Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine. Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.

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arXiv2018

Gradient estimates for a class of anisotropic nonlocal operators

Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain Hölder estimates for the solution.

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arXiv2017

A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

We consider bounded solutions of the nonlocal Allen-Cahn equation $$ (-Δ)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicity condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in which~$s\in\left(0,\frac12\right)$. Under the limit assumptions $$ \lim_{x_n\to-\infty} u(x',x_n)=-1\quad{\mbox{ and }}\quad \lim_{x_n\to+\infty} u(x',x_n)=1,$$ it has been recently shown that~$u$ is necessarily $1$D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by Ennio De Giorgi.

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arXiv2017

Density estimates for degenerate double-well potentials

We consider a general energy functional for phase coexistence models, which comprises the case of Banach norms in the gradient term plus a double-well potential. We establish density estimates for $Q$-minima. Namely, the state parameters close to both phases are proved to occupy a considerable portion of the ambient space. From this, we obtain the uniform convergence of the level sets to the limit interface in the sense of Hausdorff distance. The main novelty of these results lies in the fact that we do not assume the double-well potential to be non-degenerate in the vicinity of the minima. As far as we know, these types of density results for degenerate potentials are new even for minimizers and even in the case of semilinear equations, but our approach can comprise at the same time quasilinear equations, $Q$-minima and general energy functionals.

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arXiv2017

A sharp Bernstein-type theorem for entire minimal graphs

We consider entire solutions $u$ to the minimal surface equation in $R^N$, with $ N\ge8,$ and we prove the following sharp result : if $N-7$ partial derivatives $ \frac{\partial u }{\partial {x_j}}$ are bounded on one side (not necessarily the same), then $u$ is necessarily an affine function.

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arXiv2017

Flatness results for nonlocal minimal cones and subgraphs

We show that nonlocal minimal cones which are non-singular subgraphs outside the origin are necessarily halfspaces. The proof is based on classical ideas of~\cite{DG1} and on the computation of the linearized nonlocal mean curvature operator, which is proved to satisfy a suitable maximum principle. With this, we obtain new, and somehow simpler, proofs of the Bernstein-type results for nonlocal minimal surfaces which have been recently established in~\cite{FV}. In addition, we establish a new nonlocal Bernstein-Moser-type result which classifies Lipschitz nonlocal minimal subgraphs outside a ball.

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arXiv2017

Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips

We consider nonnegative solutions to $-Δu=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.

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arXiv2016

Regularity and rigidity theorems for a class of anisotropic nonlocal operators

We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order~$2$ in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

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arXiv2015

$f$-extremal domains in hyperbolic space

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain $Ω\subset \mathbb{H} ^2$ with $C^2$ boundary whose complement is connected and supports a bounded positive solution $u$ to an overdetermined problem, assuming natural conditions on the equation and the behaviour at infinity of the solution, must be either a geodesic ball or, a horodisk or, a half-space determined by a complete equidistant curve or, the complement of any of the above example. Moreover, in each case, the solution $u$ is invariant by the isometries fixing $Ω$.

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arXiv2015

Monotonicity in half-spaces of positive solutions to $-Δ_p u=f(u)$ in the case $p>2$

We consider weak distributional solutions to the equation $-Δ_pu=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\leq2$ is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville type theorems for the Lane-Emden type equation. Furthermore any nonnegative solution turns out to be $C^{2,α}$ smooth.

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arXiv2015

A rigidity result for nonlocal semilinear equations

We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.

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